GALOIS EQUIVARIANCE AND STABLE MOTIVIC HOMOTOPY …Hopkins [8], stable equivariant homotopy theory controls the chromatic decomposition of stable homotopy theory. It is also essential

arX

iv:1

401.

4728

v3 [

mat

h.A

T]

23

Sep

2015

GALOIS EQUIVARIANCE AND

STABLE MOTIVIC HOMOTOPY THEORY

J. HELLER AND K. ORMSBY

Abstract. For a finite Galois extension of fields L/k with Galois group G, we study afunctor from the G-equivariant stable homotopy category to the stable motivic homotopycategory over k induced by the classical Galois correspondence. We show that aftercompleting at a prime and η (the motivic Hopf map) this results in a full and faithfulembedding whenever k is real closed and L = k[i]. It is a full and faithful embeddingafter η-completion if a motivic version of Serre’s finiteness theorem is valid. We produce

strong necessary conditions on the field extension L/k for this functor to be full andfaithful. Along the way, we produce several results on the stable C2-equivariant Bettirealization functor and prove convergence theorems for the p-primary C2-equivariantAdams spectral sequence.

1. Introduction

The stable versions of equivariant and motivic homotopy theory play important rolesin the geometry of manifolds, algebraic cycles, and quadratic forms. Stable equivarianthomotopy theory is the study of topological spaces equipped with a group action up to stableequivariant weak equivalence. It has recently found stunning application [22] to the Kervaireproblem, playing an essential role in the proof that there are no smooth framed manifoldsof Kervaire invariant one in dimensions greater than 126. Via the work of Devinatz andHopkins [8], stable equivariant homotopy theory controls the chromatic decomposition ofstable homotopy theory. It is also essential to the study of topological Hochschild homology[4].

Motivic homotopy theory is a homotopy theory of schemes in which the affine line playsthe role of the unit interval. Its study was initiated by Morel and Voevodsky [40] in work re-lated to Rost and Voevodsky’s resolution of the Bloch-Kato conjectures on Milnor K-theoryand Galois cohomology [49, 52]. Its stable version plays an essential role in the theoryof motives and motivic cohomology [53]. This circle of ideas led to the resolution of theMilnor conjecture on quadratic forms [41] and the Quillen-Lichtenbaum conjecture, a pow-erful result linking algebraic K-theory and values of Dedekind ζ-functions via a “homotopylimit problem” phrased in the language of stable equivariant homotopy [15]. Stable motivichomotopy theory also opens new vistas, such as the study of algebraic cobordism [48].

The purpose of this paper is to study how equivariant and motivic stable homotopy theoryare related via the classical Galois correspondence.

A fundamental computation in stable motivic homotopy theory is the identification of theendomorphism ring of the motivic sphere spectrum by Morel [39]. In loc. cit. Morel showsthat EndSHk

(Sk) is isomorphic to the Grothendieck-Witt group GW (k) of nondegenerate

2010 Mathematics Subject Classification. Primary: 14F42, 55P91 Secondary: 11E81, 19E15.Key words and phrases. Equivariant and motivic stable homotopy theory, equivariant Betti realization.

1

http://arxiv.org/abs/1401.4728v3

2 J. HELLER AND K. ORMSBY

quadratic forms over a perfect field k. It is now a classical fact, going back to Segal and tomDieck, that the endomorphism ring EndSHG(SG) of the equivariant sphere spectrum in theequivariant stable homotopy category is equal to the Burnside ring A(G) of finite G-sets.

When L/k is a finite Galois extension with Galois group G, Dress [10, Appendix B](see also, [3, §4]) constructs a ring homomorphism A(G) → GW (k) relating these twofundamental invariants. In fact, the Galois correspondence can be stabilized to yield astrong symmetric monoidal triangulated functor from the stable G-equivariant homotopycategory to the stable motivic homotopy category over k,

c∗L/k : SHG → SHk.

This relies on work of P. Hu [26]. When L = k, c∗L/L is simply the functor induced by

sending a simplicial set to its associated constant motivic space. When L = k is algebraicallyclosed of characteristic zero, Levine [34] has recently shown that c∗L/L is a full and faithful

embedding, but this is not the general case for c∗L/k. Indeed, the Burnside ring A(G)

is always torsion free while GW (k) can in general contain torsion, which eliminates thepossibility of c∗L/k inducing an isomorphism A(G) ∼= GW (k). However, if k is a real closed

field then GW (k) and A(C2) are isomorphic so one might still hope that Levine’s embeddingtheorem can be generalized to real closed fields. Our main result, proved in Theorem 2.21and Theorem 2.22 below, is that this indeed is the case after (p, η)-completion. Here p isa prime and η is the motivic Hopf map induced by the canonical projection A2 r 0 → P1.(Details on (p, η)-completion are provided at the start of Section 2.) Moreover, the functoris a full and faithful embedding after η-completion alone if πn(Sk)Q = 0 for n > 0. Thevanishing of these higher homotopy groups would be a motivic version of the classical resultof Serre on the homotopy groups of spheres and is already known to be true when −1 is asum of squares in the basefield.

Theorem 1.1. Let k be a real closed field and L = k[i] be its algebraic closure. Then forany prime p the functor

c∗L/k : SHC2→ SHk

is a full and faithful embedding after (p, η)-completion. If πn(Sk)Q = 0 for n > 0 it is a fulland faithful embedding after η-completion.

It is a consequence of [28, Theorem 1] that (2, η)-completion is the same as 2-completionwhen k is real closed, so the above theorem specializes at p = 2 to say that c∗L/k is full and

faithful after 2-completion when k is real closed.

Remark 1.2. In order to deduce integral full faithfulness of c∗L/k from η-complete full

faithfulness, one would need to control the η-periodic (i.e., η-inverted) stable homotopycategories as well. Recent work of Guillou-Isaksen and Andrews studies the η-periodic 2-complete sphere over C from a computational perspective, but there aren’t many techniquesdeveloped for working with purely η-periodic objects in general.

1.1. Computational ramifications. Our embedding result has significant implicationsfor (Picard-graded) stable homotopy groups of spheres in the C2-equivariant and real closedmotivic settings. Recall that the representation spheres Sm+nσ are invertible in SHC2

whereSm+nσ is the one-point compactification ofm copies of the one-dimensional real trivial repre-sentation and n copies of the real sign representation. As such Z⊕Zσ is a subgroup of thePicard group of invertible objects in SHC2

, and it is common to consider the bigraded stablehomotopy groups πm+nσX = [Sm+nσ, X ]C2

of a C2-spectrum X . When k is real closed and

GALOIS EQUIVARIANCE AND STABLE MOTIVIC HOMOTOPY THEORY 3

L = k[i], Theorem 1.1 implies that c∗L/k : πm+nσ(SC2)∧p,η

∼= [c∗L/kSm+nσ, (Sk)

∧p,η]k. We will

see that c∗L/kSm+nσ ≃ Sm ∧ (SL)∧n where SL is the unreduced suspension of Spec(L). By

a theorem of P. Hu [26], SL is invertible and Z ⊕ ZL is a subgroup of the Picard groupof SHk. We emphasize that SL is not weakly equivalent to A1 r 0 and this is not the“standard” bigrading in motivic homotopy theory.

Regardless, if we set Sm+nL = Sm ∧ (SL)∧n and make the natural definition of πm+nL,we see that c∗L/k induces isomorphisms

πm+nσ(SC2)∧p,η

∼=−→ πm+nL(Sk)∧p,η

for all m,n ∈ Z under the conditions of Theorem 1.1. It is an observation of D. Dugger thatthe same result does not hold if SL is replaced by A1 r 0.

The C2-equivariant stable stems were studied by Araki and Iriye via Toda-style methods.In [2], they compute the groups πm+nσSC2

for m+ n ≤ 8. In particular, they compute thegroups πmSC2

for m ≤ 8, so Theorem 1.1 implies the following corollary.

Corollary 1.3. If k is a real closed field, then πm(Sk)∧2 , 0 ≤ m ≤ 8, is the 2-completion of

the values displayed in the following table.

m 0 1 2 3 4 5 6 7 8

πmSk Z2 (Z/2)3 (Z/2)3(Z/24)2

⊕Z/8Z/2 0 (Z/2)3

(Z/240)2⊕Z/16⊕ Z/2

(Z/2)7

In addition, in Corollary 2.24 we show that the 2-complete version of Morel’s conjectureon π1(Sk) holds for real closed fields. The integral version of this conjecture says that, fora general base field F , there is a short exact sequence

0 → KM2 (F )/24 → π1SF → F×/(F×)2 ⊕ Z/2 → 0.

The second-named author and P. Østvær have previously verified the integral version ofMorel’s conjecture for fields of cohomological dimension less than three [42].

While these immediate applications transfer information from C2-equivariant to motivichomotopy over a real closed field, future work should leverage motivic homotopy to produceC2-equivariant computations. In particular, the dual motivic Steenrod algebra is smallerthan its equivariant counterpart, making Adams and Adams-Novikov spectral sequencecomputations more approachable. The authors plan to apply these tools over the field R ofreal numbers (with the above exotic Picard grading) in order to extend our computationalunderstanding of the stable C2-equivariant homotopy category.

1.2. Galois correspondence and motivic homotopy theory. An intriguing viewpointon our embedding theorem is as a generalization of the classical Galois correspondence inthe case of real closed fields. Indeed, if L/k is a finite Galois extension with Galois group Gthen the Galois correspondence is an equivalence between the category of finite G-sets andthe category of finite etale k-algebras. Restricting to the orbit category, this correspondencegives the functor

cL/k : OrG → Sm/k

to smooth k-schemes which is explicitly given on objects by cL/k(G/H) = Spec(LH).As recorded in Theorem 4.6, this functor can be stabilized, yielding a strong symmetricmonoidal, triangulated functor

c∗L/k : SHG → SHk.


It is not hard to see that the unstable version of this functor induces a full and faithfulembedding from the unstable G-equivariant homotopy category to the unstable motivichomotopy category over k (see Lemma 4.5). Note, though, that the stable equivarianthomotopy category is formed by stabilizing with respect to representation spheres while themotivic homotopy category is formed by stabilizing with respect to P1. Hence there is noreason for this pleasant relationship between the two categories to remain after stabilization,yet it does in special cases. In fact, we can say something slightly more precise. The image ofc∗L/k is always contained in the subcategory Ek of SHk which is generated by the finite etale

k-algebras.1 Our result can thus be rephrased as an equivalence of triangulated categoriesbetween SHC2

and Ek when k is real closed.This translation of stable motivic homotopy over k into stableG-equivariant homotopy for

G = Gal(L/k) will not work for general finite Galois extensions L/k. Indeed, in Theorem 3.4we show that c∗L/k induces an isomorphism A(G) → GW (k) if and only if either k is

quadratically closed and L = k, or k is euclidean2 and L = k[i]. This implies in particularthat c∗L/k cannot be full and faithful if L/k is not of this special form.

1.3. Outline of the proof. Our main theorem is directly inspired by M. Levine’s theoremon full faithfulness of the constant presheaf functor [34], and our methods are, largely, inthe same spirit as his. That said, Levine’s arguments rely on the convergence of the slicespectral sequence, a result not yet known over fields with infinite cohomological dimension.To remedy this situation, we compare the motivic and equivariant Adams spectral sequences.

Let k be real closed and set L = k[i] so that G = C2 is cyclic of order 2. By a densityargument, to show that c∗L/k is full and faithful after η-completion, it suffices to show that

c∗L/k induces isomorphisms

[Sn ∧X,Y ]C2

∼=−→ [Sn ∧ c∗L/k(X)∧η , c∗L/k(Y )∧η ]k

where X,Y take values in the set (SC2)∧η , C2+ ∧ (SC2

)∧η . The key case is when k admitsa real embedding and in this case we can use the C2-equivariant Betti realization. Thecomputation is broken up into pieces: the (p, η)-completed sphere (for any prime p) and therationalized η-complete sphere. The computation concerning the latter object relies on theconjectural motivic version of Serre’s finiteness theorem and so the η-complete version ofthe embedding theorem is conditional upon the validity of this conjecture. Of course, thefull and faithful embedding of (p, η)-completed homotopy categories holds independent ofthis conjecture. In the (p, η)-complete case, we identify the C2-equivariant Betti realizationof the motivic Adams spectral sequence with the C2-equivariant Adams spectral sequencebased on the Bredon cohomology spectrum HZ/p. We establish an equivariant version ofSuslin-Voevodsky’s theorem on Suslin homology which implies that the realization inducesan isomorphism on weight zero components of the E1-pages from which we deduce the resultin this case.

1.4. Comments on realization and profinite Galois extensions. We conclude bymaking a few comments on the role of “realization” functors. M. Levine uses the Bettirealization functor ReB : SHL → SH for algebraically closed subfields L of C to prove hisfull faithfulness theorem in [34]. Since ReB c∗ = id, the constant presheaf functor is alwaysfaithful for any k ⊆ C. Levine’s innovation was to compare the Betti realization of the

1Here “generated” means that Ek is the smallest localizing subcategory of SHk containing all (suspensionspectra of) finite etale k-algebras.

2A field k is euclidean if −1 is not a sum of squares in k and [k×: (k×)2] = 2.


slice spectral sequence for the motivic sphere spectrum over an algebraically closed fieldwith the Novikov spectral sequence in topology. An isomorphism between the E2-termsof these spectral sequences implies an isomorphism on stable homotopy groups of sphereswhich ultimately implies the fullness result.

When k has a real embedding, then there is an associated C2-equivariant Betti realizationReC2

B : SHk → SHC2. As previously mentioned, we cannot use the slice spectral sequence

to prove our embedding theorem, but our arguments still rely on using (equivariant) Bettirealization to compare some spectral sequences (namely the motivic and equivariant Adams

spectral sequences). Again, faithfulness of c∗k[i]/k is easy because ReC2

B c∗k[i]/k = id.

Suppose L = k is the algebraic closure of k and G is the absolute Galois group Gal(L/k),which is a profinite group. A natural question is whether the main theorem of this paperextends to a full faithfulness theorem for G-equivariant stable homotopy inside of SHk. Inorder to precisely state such a question, though, one would need an appropriate notion ofgenuine G-spectra and G-stable homotopy when G is profinite. Proposals for this categoryare contained in [16, 44], and C. Barwick has communicated ideas on an alternate formu-lation to the authors. Whichever model is chosen, one would hope that it would admitwell-behaved functors

c∗L/k : SHG → SHk and ReGB : SHk → SHG

such that ReGB c∗L/k is some form of pro-completion of the identity functor. This would

result in a pro-faithfulness theorem, at which point one could examine fullness propertiesas well. The authors hope to pursue this line of inquiry in future research.

1.5. Organization of the paper. We prove our main theorem in §2 according to thestrategy outlined above. We then deduce several interesting corollaries, including our Picard-graded homotopy comparison (Corollary 2.23), Morel’s conjecture on π1Sk for real closedfields (Corollary 2.24), and a relative version of our theorem comparing full faithfulness ofc∗L/L and full faithfulness of c∗L/k (Corollary 2.26).

In §3, we study the effect of c∗L/k on the endomorphism ring of the sphere spectrum. We

show that it induces an isomorphism if and only if either k is quadratically closed and L = k,or k is euclidean and L = k[i] (Theorem 3.4); in particular this places strong conditions onL/k necessary in order for c∗L/k to be full and faithful.

We collect several technical constructions and results in §4. In §4.1 and §4.2 we recallsome definitions and facts about different model structures we use. With these preliminariesin order, the unstable and stable versions of c∗L/k are constructed in §4.3. In §4.4 we record

the construction of and some well-known results on the stable C2-equivariant Betti realiza-tion functor arising from a real embedding of fields. In §4.5 we prove basic compatibilityresults between c∗L/k and various change-of-group and change-of-base functors. Finally, in

§4.6 we study the effect of stable C2-equivariant Betti realization on motivic cohomology.In particular, we show that the Beilinson-Lichtenbaum conjectures can be rephrased forreal closed subfields of R in terms of Bredon cohomology (Theorem 4.18) and we estab-lish an equivariant version of a theorem of Suslin-Voevodsky for torsion effective motives(Theorem 4.19).

1.6. Relation to other work. It is interesting to contrast the subject of this paper withHu, Kriz, and Ormsby’s stable equivariant motivic homotopy theory [29]. In that setup onestudies smooth schemes equipped with a G-action, G a finite group. It should be emphasizedthat this group does not necessarily have any relationship with the automorphisms of a field


extension. In contrast, in the present work we study the image of the stable Gal(L/k)-equivariant homotopy category inside the stable nonequivariant motivic homotopy categoryover k. It would be interesting to combine these notions of equivariance and geometryfurther by studying (G,Gal(L/k))-homotopy inside of the G-motivic homotopy categoryover k.

1.7. Notation and conventions. Throughout k is a perfect field and L/k is a finite Galoisextension with Galois group G. For a finite group G we write SHG for the (genuine) stableequivariant homotopy category. We write Sm/k for the category of smooth schemes offinite type over a base field k and we write SHk for the stable motivic homotopy category.We use the notation [−,−]G = SHG(−,−) and [−,−]k = SHk(−,−) for morphism sets inrespective stable homotopy categories. Our indexing convention for motivic spheres is thatSa+bα := (S1)∧a ∧ (A1 r 0)∧b. When G = C2 we write Sσ for the sign-representationsphere and set Sa+bσ := (S1)∧a ∧ (Sσ)∧b. In the special case a = b = 0, we write Sk and SGfor the sphere spectra in the motivic and equivariant categories, respectively.

For the sake of typographical simplicity, we do not use any special notational device forderived functors in §2 and §3, where we only work on the level of homotopy categories. In§4 we work in both model categories and associated homotopy categories and in this sectionwe use “derived functor notation” (i.e. LF and RF respectively for left and right derivedfunctor of F ).

Acknowledgements. We are grateful to Marc Levine and Dan Dugger for spotting errorsin previous drafts of this paper. We thank Paul Arne Østvær, Kirsten Wickelgren, andthe anonymous referee for helpful comments. We have also benefitted from the AlgebraicTopology semester at MSRI in Spring 2014. The first author also thanks the MIT mathdepartment for generous hospitality during the preparation of this paper. The second authorgratefully acknowledges support from the NSF.

2. Embedding theorem

Let L/k be a Galois extension of fields with Galois group G. As mentioned in the intro-duction, the functor OrG → Sm/k which is defined on objects by G/H 7→ Spec(LH), inducesa functor c∗L/k : SHG → SHk on stable homotopy categories. Details on this construction

are given in Section 4.Our embedding result concerns certain completions of the functor c∗L/k. Recall that

the (p, η)-completion X∧p,η of a motivic spectrum is defined to be the Bousfield localiza-

tion of X at Sk/(p, η) := cofiber(Sα ∧ Sk/p → Sk/p). We have a motivic equivalence(Sk)

∧p,η ≃ holimSk/(p

n, ηn). Similarly, for a C2-spectrum Y , define Y ∧p,η to be the Bous-

field localization of Y at the spectrum SC2/(p, η).3 We have an equivariant equivalence

(SC2)∧p,η = holim SC2

/(pn, ηn). Write (SHk)∧p,η ⊆ SHk and (SHC2

)∧p,η ⊆ SHC2respectively

for the full subcategories of (p, η)-complete objects. Note that these are triangulated sub-categories. Write (c∗L/k)

∧p,η := (−)∧p,η c∗L/k. In this section we prove that if k is a real closed

field and L = k[i] then(c∗L/k)

∧p,η : (SHC2

)∧p,η → (SHk)∧p,η

is a full and faithful embedding for any prime p. Additionally if πn(Sk)Q = 0 for any n > 0(see Conjecture 2.13) then the functor

(c∗L/k)∧η : (SHC2

)∧η → (SHk)∧η

3The map η : Sσ → S0 in SHC2is the stable map induced by C2 − 0 → CP 2.


is full and faithful without p-completion. This is proved in Theorem 2.21 and Theorem 2.22.The main step is to show that the C2-equivariant Betti realization induces isomorphisms

(i) ReC2

B,φ : [Sn, (Sk)∧p,η]k

∼=−→ [Sn, (SC2)∧p,η]C2

, and

(ii) ReC2

B,φ : [Spec(L)+ ∧ Sn, (Sk)∧p,η]k∼=−→ [C2+ ∧ Sn, (SC2

)∧p,η]C2

whenever there is a real embedding φ : k → R.

2.1. Completing at p and η. Let p be a prime. We analyze the image, under equivariantBetti realization, of the motivic Adams spectral sequence over a real closed subfield of R.Let HZ/p denote the mod-p motivic cohomology spectrum. The motivic Adams spectralsequence for Sk arises as the totalization spectral sequence of the semi-cosimplicial P1-spectrum with s-th spectrum (HZ/p)∧s and co-face maps induced by the unit Sk → HZ/p.We use the following specialization of a theorem of P. Hu, I. Kriz, and the second author.

Theorem 2.1 ([28, Theorem 1]). Let k be a real closed field, L = k[i], and let Y be eitherSk or Spec(L)+. The motivic Adams spectral sequence

Es,t1 = [St ∧ Y, (HZ/p)∧s]k =⇒ [St−s ∧ Y, (Sk)∧p,η]k.is strongly convergent. If p = 2 then [St−s ∧ Y, (Sk)∧2,η]k = [St−s ∧ Y, (Sk)∧2 ]k.Proof. This is the weight zero portion of the p-primary motivic Adams spectral sequenceconstructed in [28] over k (when Y = Sk) or over L (when Y = Spec(L)+). The form ofthe E1-page is immediate from the totalization construction. Convergence follows from [28,Theorem 1], which states that over a field k of characteristic 0, the Adams spectral sequencefor a finite cell spectrum at p converges to (p, η)-completions. Moreover, by loc. cit. ifcd2(k[i]) < ∞ then (Sk)

∧2 → (Sk)

∧2,η induces an isomorphism on motivic homotopy groups.

Real closed fields satisfy cd2(k[i]) <∞ and so we can indeed invoke [28, Theorem 1].

We now turn to the C2-equivariant Adams spectral sequence. This spectral sequence hashas been studied for p = 2 by P. Hu and I. Kriz [27], where it is shown that it convergesto the 2-completion. For odd p, the situation is a little different. The target of this spec-tral sequence is the HZ/p-nilpotent completion, which can be different than p-completion.We briefly recall its definition and construction and then show that it agrees with (p, η)-completion in general.

Bousfield’s construction and discussion of the nilpotent completion and its relation tothe Adams spectral sequence in [5] applies as well to the equivariant setting. For a conciserecollection, see [14, Section 6.7] (the discussion of loc. cit. is tailored to the motivic settingbut applies to the equivariant setting with evident modification). Let E be a C2-equivariantring spectrum and define E to be the fiber of the unit map SC2

→ E. For a spectrum X wedefine

Xs = E∧s ∧X and Cs = cofiber(Xs+1 → X).

There are maps Xs+1 → Xs, and hence maps Cs → Cs−1 induced by E → SC2. The

E-nilpotent completion of X is defined to be

X∧E = holim(Cs).

Note that there are cofiber sequences X∞ → X → X∧E , where X∞ := holimXs. The tower

Cs forms an E-nilpotent resolution of X , in the sense of [5, Definition 5.6]. The Tot-towerassociated to the cosimplicial spectrum E∧• ∧X also forms an E-nilpotent resolution of X .The arguments of [5, Proposition 5.8] thus show that the homotopy limit of this Tot-toweris homotopic to X∧

E .


Set Ws := E ∧ Xs = E ∧ E∧s ∧ X , then Ws = cofiber(Xs+1 → Xs). Note that we

also have that ΣWs = cofiber(Cs → Cs−1). By induction, each Cs is E-local and henceso is X∧

E . Therefore the map α : X → X∧E factors through the Bousfield localization,

X → LEX → X∧E .

Lemma 2.2 ([5]). The map β : LEX → X∧E is an equivariant weak equivalence if and only

if α∧E : X∧

E → (X∧E)

∧E is an equivariant weak equivalence.

Proof. This is similar to [5, p. 273]. We have a retraction E ∧ X → E ∧ X∧E → E ∧ X

obtained from X∧E → C0 = E ∧ X together with E ∧ E → E. One finds that X → Y

is an E-equivalence if and only if X∧E → Y ∧

E is an equivariant equivalence. In particular,X∧E → (LEX)∧E is an equivariant equivalence. The map β is an equivalence if and only if it

is an E-equivalence and so β is an equivalence if and only if (LEX)∧E ≃ X∧E . This happens

if and only if α∧E is an equivalence.

Proposition 2.3. Let R be a subring of Q. Suppose that E satisfies the condition that thegeometric fixed points spectrum ΦK(E), K = e, C2 are NK-connective for some NK , andHr(Φ

K(E)) is a finitely generated R-module for all r. Let X be an C2-spectrum such that

each ΦK(X) is MK-connective for some MK. Then α∧E : X∧

E≃−→ (X∧

E)∧E is an equivariant

equivalence.

Proof. There are functorial cofiber sequences X∞ → X → X∧E . The result follows by

showing that (X∞)∞ ≃ X∞. We claim that the map

(X∞)s = E∧s ∧ holim

n(E

∧n ∧X) → holimn

E∧s+n ∧X

is an equivariant weak equivalence. There is a cofiber sequence holimi Yi →∏i Yi →

∏i Yi,

and so it suffices to see that E∧s ∧ ∏

n(E∧n ∧ X) → ∏

nE∧s+n ∧ X is an equivariant

weak equivalence. It follows from [1, Thereom III.15.2] and Lemma 2.4 that this map is aweak equivalence on geometric fixed points as well as on the underlying spectrum and soit is an equivariant weak equivalence. The map (X∞)s → X∞ is thus an equivariant weakequivalence and so taking homotopy limits we have that (X∞)∞ ≃ X∞ as desired.

For a C2-spectrum E, we write πC2n (E) = [Sn, E]C2

for the nth stable equivariant homo-topy group.

Lemma 2.4. Let Yi, i ∈ N, be C2-spectra. Suppose that there is an integer N so that theunderlying spectrum of Yi is N -connective. Then ΦC2(

∏i Yi) ≃

∏i Φ

C2(Yi).

Proof. The geometric fixed points of X are equal to (EC2 ∧ X)C2 . We have an equivari-

ant equivalence EC2 ≃ colimk Skσ. Note that Skσ ∧ (

∏Yi) ≃

∏(Skσ ∧ Yi), since S

kσ isdualizable. We thus need to see that the map

(2.5) colimk

∏(Skσ ∧ Yi) →

∏colimk

(Skσ ∧ Yi)

induces an isomorphism on πC2n for all n. Consider the cofiber sequence

C2+ ∧ Skσ ∧ Yi → Skσ ∧ Yi → S(k+1)σ ∧ Yi.Since each Yi isN -connective, for a fixed n, there is an integer s such that πC2

n (C2+∧Skσ∧Yi)vanishes for all k > s. This implies that πC2

n (Skσ ∧ Yi) = πC2n (S(k+1)σ ∧ Yi). We thus have

that the (2.5) induces an isomorphism on πC2n , as desired.


Recall that πn(X) denotes the Mackey functor homtopy groups. Say that a C2-spectrumX is n-connective πk(X) = 0 for k < n. Say that a map f : X → Y is an n-equivalence ifits cofiber is n + 1-connective. For the following lemma, note that since η is zero on HZ,the map HZ → HZ/η is split.

Lemma 2.6. Let X be a connective C2-spectrum and p an odd prime. Then the unit mapX/(ps, ηt) → HZ ∧X/(ps, ηt) and the map X/(2s, ηt) → HZ ∧X/2s are 1-equivalences forany integer s, t ≥ 1.

Proof. There are cofiber sequences X/(p, ηt) → X/(ps, ηt) → X/(ps−1, ηt) and a similar onefor quotients by powers of η. An inductive argument shows that it suffices to consider thecase s = 1, t = 1. It suffices to consider X = SC2

, in which case a straightforward calculationshows that SC2

/(p, η) → HZ∧SC2/(p, η) and SC2

/(2, η) → HZ∧SC2/2 are 1-equivalences.

Proposition 2.7. Let X be a connective, C2-spectrum and p a prime. Suppose that Xsatisfies the condition that both multiplication by ps and by ηt are equal to zero on X, forsome integers s, t ≥ 1. Then X → X∧

HZ/p is an equivariant equivalence.

Proof. We treat the case of an odd prime explicitly, p = 2 is similar. First note that if ps

and ηt both act by zero on a spectrum Z, then Z is a summand of Z/(ps, ηt). Note as wellthat the previous lemma implies that if Y is n-connective, for n ≥ 0, then Y → HZ ∧ Y isan n+ 1-equivalence. We inductively define cofiber sequences Xi+1 → Xi → Ki by lettingX0 := X and Ki := HZ ∧ Xi. We claim that Xi is i-connective for all i. Indeed if Xi isi-connective then Xi/(p

s, ηt) → Ki/(ps, ηt) is an i + 1-equivalence. But this map contains

Xi → Ki as a summand and so it is an i+ 1-equivalence as well which implies that Xi+1 isi+ 1-connective.

Write Ci = cofiber(Xi+1 → X). The tower Ci is an HZ-nilpotent resolution of Xand we claim that it is in fact an HZ/p-nilpotent resolution. If N is HZ/p-nilpotent, thencolimi[Y ∧ Xi, N ]C2

= 0, where Y = S0 or C2. It remains to see that the Ci are HZ/p-nilpotent. This is seen by induction by noting the Ki are HZ/p-nilpotent since there is asplitting of Ki → HZ/p ∧Ki as follows. We have HZ/pN ∧Ki = (HZ ∧Xi) ∨ (ΣHZ ∧Xi)and so a splitting of Ki → HZ/p ∧Ki is obtained via the composition

HZ/p ∧Ki → HZ/pN ∧Ki → HZ ∧Xi = Ki.

SinceXi is i-connective, we have that holimiXi ≃ ∗ and thereforeX = X∧HZ/p as desired.

Proposition 2.8. Let X be a connective C2-spectrum and p a prime. Then there is anatural equivariant equivalence X∧

HZ/p ≃ X∧p,η.

Proof. The map X → X∧p,η is an SC2

/(p, η)-equivalence, and therefore an HZ/p-equivalence.It follows that X∧

HZ/p → (X∧p,η)

∧HZ/p is an equivariant weak equivalence. On the other hand,

Proposition 2.7 implies that X/(pn, ηn) → (X/(pn, ηn))∧HZ/p is an equivariant equivalence

for all n. Therefore we have that X∧p,η → holimn(X/(p

n, ηn))∧HZ/p ≃ (X∧p,η)

∧HZ/p.

Lemma 2.9 ([27, Corollary 6.47]). Let X be a connective C2-spectrum. There is a naturalequivariant equivalence X∧

HZ/2 ≃ X∧2 .

Proof. By the previous proposition it suffices to show that i : X∧2 → X∧

2,η is an equivariantequivalence. The map i is an equivalence after forgetting the action, so it suffices to showthat it induces an isomorphism on πC2

n . Write F for the homotopy fiber of i. Note thatη : Sσ ∧ F → F is an equivariant equivalence. Note as well that ρ : F → Sσ ∧ F is a weakequivalence, since F is nonequivariantly contractible. We have the relation η2ρ = −2η. In


particular, we find that 2 is an equivalence on F and so F/2s ≃ ∗ for all s. Since F is2-complete, we have F ≃ ∗.

Fix an embedding φ : k → R and consider the resulting C2-equivariant Betti realiza-tion ReC2

B,φ : SHk → SHC2(see Section 4.4 for details). By Theorem 4.17, the equivariant

Betti realization takes the motivic cohomology spectrum HZ/p to the Bredon cohomology

spectrum HZ/p associated to the constant Mackey functor Z/p. Since ReC2

B,φ is symmetric

monoidal and takes the unit for HZ/p to the unit for HZ/p, we see that ReC2

B,φ takes the

semi-cosimplicial P1-spectrum (HZ/p)∧• to the semi-cosimplicial C2-spectrum (HZ/p)∧•.The totalization spectral sequence for this latter object is the C2-equivariant Adams spec-tral sequence, which has been studied by P. Hu and I. Kriz [27] and the case p = 2 of thefollowing theorem is [27, Corollary 6.47].

Theorem 2.10. Let Y be either SC2or C2+. The C2-equivariant Adams spectral sequence

Es,t1 = [St ∧ Y, (HZ/p)∧s]C2=⇒ [St−s ∧ Y, (SC2

)∧p,η]C2

is strongly convergent. If p = 2 then (SC2)∧2,η = (SC2

)∧2 .

Proof. The spectral sequence associated to the Tot-tower of the semi-cosimplicial object

Y ∧ (HZ/p)∧s agrees with the spectral sequence associated to the tower Y ∧HZ/p∧s. This

in turn agrees with the spectral sequence with Y replaced by Y ∧HZ/p. This spectral sequence

converges to Y ∧HZ/p since we have that holims Y

∧HZ/p∧HZ/p

∧s ≃ ∗ (as Y ∧HZ/p ≃ (Y ∧

HZ/p)∧HZ/p).

Together with the identifications of Proposition 2.8 and Lemma 2.9, this establishes theresult.

By comparing these two Adams spectral sequences, we obtain the following result.

Proposition 2.11. Let k be real closed, set L = k[i], and let φ : k → R be an embedding offields. Then the induced maps

(i) ReC2

B,φ : [Sn, (Sk)∧p,η]k

∼=−→ [Sn, (SC2)∧p,η]C2

, and

(ii) ReC2

B,φ : [Spec(L)+ ∧ Sn, (Sk)∧p,η]k∼=−→ [C2+ ∧ Sn, (SC2

)∧p,η]C2

are isomorphisms for any n ∈ Z. For p = 2, the induced maps [Sn, (Sk)∧2 ]k

∼=−→ [Sn, (SC2)∧2 ]C2

and [Spec(L)+ ∧ Sn, (Sk)∧2 ]k∼=−→ [C2+ ∧ Sn, (SC2

)∧2 ]C2are isomorphisms.

Proof. We have already noted that ReC2

B,φ(HZ/p)∧s ≃ HZ/p∧s, and that we have a map of

Adams spectral sequences. The computation of the motivic Steenrod algebra [50, 52] showsthat we have a decomposition HZ/p∧HZ/p ≃ ∨Σpi+qiαHZ/p for appropriate (pi, qi) which inparticular satisfy qi ≥ 0. It follows from Theorem 4.19 that the equivariant Betti realizationinduces an isomorphism on the weight zero E1-page of the Adams spectral sequences. ByTheorem 2.1 and Theorem 2.10, the proposition follows.

2.2. Rational homotopy groups. For a (motivic or equivariant) spectrum X we writeXQ for the Bousfield localization at MQ, the rational Moore spectrum. If Y is a compactspectrum, then [Y,XQ] = [Y,X ]⊗Q.

The homotopy groups of the equivariant rational sphere spectrum are rather simple.

Proposition 2.12. The homotopy groups of the rational C2-sphere are π0(SC2)Q = Q⊕Q

and πn(SC2)Q = 0 for any integer n 6= 0.


Proof. This follows immediately from the well known fact (see e.g. [20, Corollary A.6]) thatfor any finite group, (SG)Q is weakly equivalent to HAQ, the Eilenberg-MacLane spectrumassociated to the rational Burnside Mackey functor.

Conjecturally the higher homotopy groups of the sphere also vanish.

Conjecture 2.13 (Motivic Serre finiteness). Let k be a field. Then πn(Sk)Q = 0 for n > 0.

Definition 2.14. We say that a field k has motivic Serre finiteness if Conjecture 2.13 holdsover k.

Rationally (in fact already when 2 is inverted) there are orthogonal idempotents ǫ+ =(ǫ − 1)/2 and ǫ− = (ǫ + 1)/2 acting on (Sk)Q, obtained from ǫ ∈ π0Sk.

4 We thus obtain arational decomposition of the sphere spectrum (Sk)Q = (Sk)

+Q ∨ (Sk)

−Q in which the factors

correspond respectively to inverting ǫ+ and ǫ−. It follows from Morel’s description [38] of(Sk)

+Q as the rational motivic cohomology spectrum HQ (see [6, Theorem 16.2.13]) that k has

motivic Serre finiteness whenever −1 is sum of squares in k (in which case (Sk)−Q vanishes).

Morel [38] also conjectures a description of (Sk)−Q which would imply that motivic Serre

finiteness holds in general.

Proposition 2.15. Let k be a real closed field, set L = k[i], and let φ : k → R be anembedding. Assume that k has motivic Serre finiteness. Then the maps

(i) ReC2

B,φ : [Sn, (Sk)Q]k∼=−→ [Sn, (SC2

)Q]C2, and

(ii) ReC2

B,φ : [Spec(L)+ ∧ Sn, (Sk)Q]k∼=−→ [C2+ ∧ Sn, (SC2

)Q]C2

are isomorphisms for any n ∈ Z.

Proof. Since ReC2

B,φ c∗L/k = id, we know that the map of the proposition is surjective. Since

GW (k) = Z⊕ Z and GW (L) = Z for any real closed field k, it follows that the first map isan isomorphism in degree zero. By the previous propositions, these groups are zero in allother degrees.

Write ǫ ∈ πC2

0 (S0) for the stable map induced by the permutation Sσ ∧ Sσ → Sσ ∧ Sσ.As in the motivic setting, once 2 is invertible there are idempotents ǫ+ = (ǫ − 1)/2 andǫ− = (ǫ + 1)/2 that induce a splitting SC2

[1/2] = SC2[1/2]+ ∨ SC2

[1/2]−.

Lemma 2.16. Let k be a field and X and object of SHk. Then (X [1/2])∧η = (X [1/2])+.

Similarly if W is a C2-spectrum, then (W [1/2])∧η =W [1/2]+.

Proof. We have that (X [1/2])∧η = (X [1/2]+)∧η ∨ (X [1/2]−)∧η . From the relation ǫη = η,

we find that η : Sα ∧ X [1/2]+ → X [1/2]+ is zero and hence X [1/2]+ is η-complete. Onthe other hand η : Sα ∧ X [1/2]− → X [1/2]− is an equivalence and so (X [1/2]−)∧η ≃ ∗. It

follows that (X [1/2])∧η = (X [1/2])+ as desired. A similar analysis applies in the equivariantsetting.

Corollary 2.17. Let k be a real closed field, set L = k[i], and let φ : k → R be an embedding.

If X is in SHk and satisfies the condition that ReC2

B,φ : πn(XQ) → πn(ReC2

B,φ(XQ)) is an

isomorphism, then ReC2

B,φ : πn((XQ)∧η ) → πn(Re

C2

B,φ(XQ)∧η ) is also an isomorphism.

4Recall that ǫ is the stable map induced by the permutation A1 r 0 ∧A1 r 0 ∼= A1 r 0 ∧A1 r 0.


Proof. The map ReC2

B,φ : πn(XQ) → πn(ReC2

B,φ(XQ)) is a direct sum of maps

(ReC2

B,φ)+ ⊕ (ReC2

B,φ)− : πn(X

+Q )⊕ πn(X

−Q ) → πn(Re

C2

B,φ(XQ)+)⊕ πn(Re

C2

B,φ(XQ)−).

The result will thus follows from the previous lemma.

2.3. Full and faithful embedding. We now assemble the previous computations to de-duce our main theorem.

Proposition 2.18. Let k be a real closed field, set L = k[i], and let φ : k → R be anembedding. Assume that k has motivic Serre finiteness. Then

(i) ReC2

B,φ : [Sn, (Sk)∧η ]k

∼=−→ [Sn, (SC2)∧η ]C2

, and

(ii) ReC2

B,φ : [Spec(L)+ ∧ Sn, (Sk)∧η ]k∼=−→ [C2+ ∧ Sn, (SC2

)∧η ]C2

are isomorphisms for all n ∈ Z.

Proof. By [42, Appendix A] there is a homotopy cartesian square in SHk

Sk //

∏p(Sk)

∧p

(Sk)Q // ∏

p((Sk)∧p )Q

where the products are over prime integers p. There is a similar equivariant arithmeticfracture square in SHC2

5. Taking the η-completion of this square yields the homotopycartesian square

(Sk)∧η

//

∏p(Sk)

∧p,η

[(Sk)Q]

∧η

// [∏p((Sk)

∧p )Q]

∧η

and similarly in SHC2. Since XQ → (X∧

η )Q is a filtered colimit of Sk/η-equivalences, it isitself an Sk/η-equivalence. It follows that [(Sk)

∧p )Q]

∧η ≃ [(Sk)

∧p,η)Q]

∧η and similarly for the C2-

equivariant case. The square obtained by applying ReC2

B,φ to the above square maps to theequivariant arithmetic fracture square. We thus obtain a comparison diagram of associatedlong exact sequences. The proposition thus follows from Proposition 2.11, Proposition 2.15,Corollary 2.17 and the five lemma.

We now turn our attention to (c∗L/k)∧η . Write ηL for the map c∗L/k(η) : S

L → S0.

Lemma 2.19. Let k be a real closed field and let L = k[i]. Then the canonical mapc∗L/k(SC2

)∧η → (Sk)∧ηL is also an equivalence. The canonical map (Sk)

∧η → (Sk)

∧η,ηL is an

equivalence. In particular (c∗L/k)∧η ((SC2

)∧η ) ≃ (Sk)∧η .

Proof. We show that the first equivalence holds for the 2-complete sphere and for spectraon which 2 is invertible. A comparison of fracture squares then implies the result. Firstnote that c∗L/k((SC2

)∧2,η) = (Sk)∧2 by Lemma 2.9 and Proposition 2.8. The map ηL induces

the HZ/2-module map ηL : SL ∧ HZ/2 → HZ/2. The group of HZ/2-module maps fromSL ∧ HZ/2 to HZ/2 is identified with the group [SL,HZ/2]k = 0. Thus ηL acts by zero on

5The authors do not know a handy reference for this equivariant arithmetic fracture square, but standardtechniques adapt to produce it. For instance, the proof giving the motivic arithmetic fracture square in [42,Appendix A] works almost verbatim in the equivariant setting.


any HZ/2-module and so any HZ/2-module is ηL-complete. It follows that (Sk)∧HZ/2 is ηL

complete which by Theorem 2.10 implies that (Sk)∧2 is ηL-complete. Now if 2 is invertible

on X then we have X∧η = X [ǫ−1

+ ] and c∗L/k(X)∧ηL = c∗L/k(X)[(ǫL)−1+ ] by Lemma 2.16. Since

c∗L/k(X [ǫ−1+ ]) = c∗L/k(X)[(ǫL)

−1+ ], we have established the first equivalence.

For the second equivalence, we compare the applications of (−)∧η and (−)∧η,ηL to the

arithmetic fracture square. Since (Sk)∧p,η ≃ (Sk)

∧HZ/p and [SL,HZ/p]k = 0 we find that

(Sk)∧p,η is ηL-complete. Let X be an object of SHk. By Lemma 2.16 we have (XQ)

∧η ≃ X+

Q

and by [6, Theorem 16.2.13] we have X+Q ≃ X ∧ HQ. Since [SL,HQ]k = 0, we find that

(XQ)∧η is ηL-complete. This implies the second equivalence.

We now convert our analysis of ReC2

B,φ to c∗L/k using a limit argument which is a modifi-

cation of the one used in [34, Lemma 6.6] to the case of real closed fields.

Proposition 2.20. Let k be a real closed field and set L = k[i]. Assume that k has motivicSerre finiteness. Then for any n ∈ Z, the maps

(i) (c∗L/k)∧η : [Sn, (SC2

)∧η ]C2

∼=−→ [Sn, (c∗L/k)∧η ((SC2

)∧η )]k, and

(ii) (c∗L/k)∧η : [C2+ ∧ Sn, (SC2

)∧η ]C2

∼=−→ [Spec(L)+ ∧ Sn, (c∗L/k)∧η ((SC2)∧η )]k

are isomorphisms. For any prime p, the maps [Sn, (SC2)∧p,η]C2

∼=−→ [Sn, (c∗L/k)∧p,η((SC2

)∧p,η)]k,

and [C2+ ∧ Sn, (SC2)∧p,η]C2

∼=−→ [Spec(L)+ ∧ Sn, (c∗L/k)∧p,η((SC2

)∧p,η)]k for (p, η)-completed

spheres are always isomorphisms. For p = 2, the maps [Sn, (SC2)∧2 ]C2

∼=−→ [Sn, (Sk)∧2 ]k, and

[C2+ ∧ Sn, (SC2)∧2 ]C2

∼=−→ [Spec(L)+ ∧ Sn, (Sk)∧2 )]k are isomorphisms.

Proof. If there is an embedding φ : k ⊆ R, then this is a direct consequence of Proposition 2.11,Proposition 2.18, and Lemma 2.19 and the relation ReC2

B,φ c∗L/k ∼= id. We treat the case of

the η-completed spheres below, the case of (p, η)-completion holds verbatim.As k is real closed, L is algebraically closed. We may express L as the union

⋃α∈A Lα

of algebraically closed subfields Lα ⊂ L of finite transcendence degree over Q indexed by awell-ordered set A. Consider the fields kα = Lα ∩ k. We claim that the kα are isomorphicto real closed subfields of R. If this is the case, then

colimα

[Sn, (c∗Lα/kα)∧η ((SC2

)∧η )]kα and colimα

[Spec(Lα)+ ∧ Sn, (c∗Lα/kα)∧η ((SC2

)∧η )]kα

are colimits of abelian groups with constant values [Sn, (SC2)∧η ]C2

and [C2+ ∧Sn, (SC2)∧η ]C2

respectively, by the observation in the first paragraph. Since it is obvious that k =⋃α kα,

using essentially smooth base change [24, Lemma A.7] we conclude that these colimits arerespectively isomorphic to [Sn, (c∗L/k)

∧η ((SC2

)∧η )]k and [Spec(L)+ ∧ Sn, (c∗L/k)∧η ((SC2

)∧η )]k.

Thus we may now conclude that the maps [Sn, (SC2)∧η ]C2

→ [Sn, (c∗L/k)∧η ((SC2

)∧η )]k and

[C2+ ∧ Sn, (SC2)∧η ]C2

→ [Spec(L)+ ∧ Sn, (c∗L/k)∧η ((SC2

)∧η )]k are isomorphisms for all real

closed fields.It remains to verify the claim that each kα is isomorphic to a real closed subfield of R.

Since Lα is algebraically closed and [Lα : kα] = 2, the Artin-Schreier theorem implies thatkα is real closed. Fix kα and choose a transcendence basis x1, . . . , xn of kα over Q in whicheach xi is positive in kα. By sending each xi to a positive transcendental real number, weproduce an order embedding of Q(x1, . . . , xn) into R. Since kα/Q(x1, . . . , xn) is a unionof finite extensions of ordered fields, [33, Proposition VIII.2.16] implies that there is anembedding kα → R, as desired.


We are now ready to prove our main theorem. Recall that a localizing subcategory E of atriangulated category T is a full triangulated subcategory, containing all direct summandsof its objects and closed under arbitrary coproducts.

Theorem 2.21. Let k be a real closed field and let L = k[i] be its algebraic closure. Assumethat k has motivic Serre finiteness. Then

(c∗L/k)∧η : (SHC2

)∧η → (SHk)∧η

is a full and faithful embedding.

Proof. Consider the subcategory C ⊆ (SHC2)∧η whose objects are η-complete C2-equivariant

spectraX such that (c∗L/k)∧η : [Sn, X ]C2

→ [Sn, c∗L/k(X)∧η ]k and (c∗L/k)∧η : [C2+∧Sn, X ]C2

→[C2+ ∧ Sn, c∗L/k(X)∧η ]k are isomorphisms for all n. This is a localizing subcategory and by

Proposition 2.20 it contains (SC2)∧η and we argue below that C2+ ∧ (SC2

)∧η is in C as well.This implies that C = (SHC2

)∧η , as this is the smallest localizing subcategory containing(SC2

)∧η , C2+ ∧ (SC2)∧η .

Now we show that C2+∧(SC2)∧η is also in C. Since c∗L/k is strong symmetric monoidal and

C2+ is dualizable, [17, Proposition 3.12] implies that for any C2-spectrum X , the naturalmap c∗L/k(F (C2+, X)) → F (Spec(L)+, c

∗L/k(X)) is an isomorphism in SHk, where F (−,−)

denotes the function spectrum in the corresponding homotopy category. Now C2+ is selfdual, i.e. there is an isomorphism C2+

∼= D(C2+) in SHC2where D(−) = F (−, SC2

) denotesthe Spanier-Whitehead dual. As with any dualizable object, there is a natural isomorphismν : D(C2+) ∧ X ∼= F (C2+, X). Combining these isomorphisms yields the isomorphismω : C2+∧X ∼= F (C2+, X) in SHC2

, which is a simple case of the Wirthmuller isomorphism,and c∗L/k(ω) induces an isomorphism Spec(L)+ ∧ c∗L/k(X)∧η

∼= F (Spec(L)+, c∗L/k(X)∧η ) in

SHk. This isomorphism together with Proposition 2.20 now implies that the maps

(i) (c∗L/k)∧η : [Sn, C2+ ∧ (SC2

)∧η ]C2→ [Sn, Spec(L)+ ∧ c∗L/k((SC2

)∧η )∧η ]k, and

(ii) (c∗L/k)∧η : [C2+∧Sn, C2+∧(SC2

)∧η ]C2→ [Spec(L)+∧Sn, Spec(L)+∧c∗L/k((SC2

)∧η )∧η ]k

are isomorphisms for any n ∈ Z.Now, for any η-complete C2-spectrumX , let LX denote the full subcategory of η-complete

C2-spectra Y such that [Sn ∧ Y,X ]C2→ [Sn ∧ c∗L/k(Y )∧η , c

∗L/k(X)∧η ]k is an isomorphism for

all n ∈ Z. It is clear that LX is a localizing subcategory of SHC2. We have seen that LX

contains both (SC2)∧η and C2+ ∧ (SC2

)∧η . Therefore LX = (SHC2)∧η . Since X was arbitrary,

we have proved that (c∗L/k)∧η is full and faithful.

Indepedent of whether k has motivic Serre finiteness, the argument in the previous the-orem yields the embedding theorem for the (p, η)-complete homotopy categories.

Theorem 2.22. Let k be a real closed field and let L = k[i] be its algebraic closure. Thenfor any prime p

c∗L/k : (SHC2)∧p,η → (SHk)

∧p,η

is a full and faithful embedding. For p = 2, c∗L/k : (SHC2)∧2 → (SHk)

∧2 is full and faithful.

As mentioned in the introduction, our main theorem has the following corollary on Picard-graded stable homotopy groups.

Corollary 2.23. Suppose k is real closed and L = k[i] and let SL denote the unreducedsuspension of Spec(L). Then for all m,n ∈ Z and any (p, η)-complete C2-spectrum X, the


functor c∗L/k induces an isomorphism of Picard-graded stable homotopy groups

πm+nσ(X)∼=−→ πm+nL(c

∗L/k(X)∧η ).

If k has motivic Serre finiteness, then it is an isomorphism for any C2-spectrum X. Inparticular, in this case

πm+nσ((SC2)∧η ))

∼=−→ πm+nL((Sk)∧η ).

We can also deduce a 2-complete version of Morel’s conjecture on π1Sk for k a real closedfield. Recall that for a general field k, Morel’s conjecture states that there is a short exactsequence

0 → KM2 (k)/24 → π1Sk → KM

1 (k)/2⊕ Z/2 → 0

in which the map π1Sk → KM1 (k)/2⊕Z/2 is induced by the unit map Sk → KO to Hermitian

K-theory and KM2 (k)/24 → π1Sk takes symbols [a, b] to [a, b]ν, ν the motivic quaternionic

Hopf map.

Corollary 2.24. If k is real closed, then π1(Sk)∧2 sits in the short exact sequence

0 → KM2 (k)/8 → π1(Sk)

∧2 → KM

1 (k)/2⊕ Z/2 → 0.

Proof. By [2], we have π1SC2= (Z/2)3 with basis ηs, [C2/e]ηs, e

2νC2where [C2/e] is the

class of C2/e in A(C2), e is represented by the canonical map S0 → Sσ, and νC2is the

C2-equivariant quaternionic Hopf map. By Corollary 2.23, there is an abstract isomorphismπ1(Sk)

∧2∼= (Z/2)3. (Recall that (2, η)-completion is the same as 2-completion when the 2-

primary cohomological dimension of k[i] is finite.) By [42, Lemma 5.12], the map π1(Sk)∧2 →

KM1 (k)/2 ⊕ Z/2 is surjective, taking 〈u〉ηs to ([u], 1) (where 〈u〉 represents the quadratic

form uX2 in GW (k)). It follows that ηs and 〈−1〉ηs are linearly independent. The C2-Bettirealization of ρ2ν is e2νC2

6= 0, and ν = 0 ∈ π1+2αKO = 0, so ρ2ν is nonzero and linearlyindependent of ηs, 〈−1〉ηs. The corollary follows.

Remark 2.25. If k is real closed, the map π1SC2→ π1Sk is given by

ηs 7→ ηs, [C2/e]ηs 7→ 〈1,−1〉ηs, e2νC27→ ρ2ν.

Finally we note that an equivariant embedding theorem implies a nonequivariant embed-ding theorem.

Corollary 2.26. Let L/k be a finite Galois extension with Galois group G. If the functorc∗L/k : SHG → SHk is full and faithful, then the constant presheaf functor c∗L/L : SH → SHLis full and faithful as well.

Proof. Assume that c∗L/k is full and faithful and consider the commutative diagram

[G+ ∧ Sn, X ]G

∼=

c∗L/k

∼=// [c∗L/k(G+ ∧ Sn), c∗L/kX ]k

∼=

[Sn, resX ]e

c∗L/L

// [c∗Sn, c∗resX ]L,

obtained using Proposition 4.12. The vertical arrows are isomorphisms, and the top hor-izontal arrow is an isomorphism by assumption. Thus the bottom horizontal arrow is anisomorphism as well. Since every spectrum is the restriction resX of some G-spectrum X ,we can use a density argument as in the proof of Theorem 2.21 to conclude that c∗L/L is full

and faithful.


3. The trace homomorphism and necessary conditions for full-faithfulness

In this section, we discuss the possibility of c∗L/k being full and faithful for more general

Galois extensions L/k. As noted in the introduction, presence of torsion in the Grothendieck-Witt group is the first obvious obstruction to an isomorphism on π0 and therefore to c∗L/kinducing a full and faithful embedding. However, there are many fields whose Grothendieck-Witt group is torsion-free and we are able to place strong restrictions on which fields k andL can have the property that c∗L/k induces an isomorphism on π0.

Recall that the classical map hL/k : A(G) → GW (k) mentioned in the introduction is theunique ring homomorphism with the property that G/H ∈ A(G) is mapped to trLH/k(〈1〉).(See [3, §4] for the basic properties of hL/k.) The functor c∗L/k : SHG → SHk also induces

a map c∗L/k : A(G) → GW (k). The following is essentially a rephrasing of M. Hoyois’s [25]

computation of the motivic Euler characteristic of a separable field extension.

Proposition 3.1. The maps c∗L/k : A(G) → GW (k) and hL/k are equal.

Proof. The identification A(G) ∼= EndSHG(SG) is given by sending a finite G-set M toits Euler characteristic χ(M) (for a recollection of Euler characteristics and their proper-ties see, e.g., [37]). The functor c∗L/k is strong symmetric monoidal and so we have that

c∗L/kχ(G/H) = χ(c∗L/k(G/H)) = χ(Spec(LH)) in EndSHk(Sk). But by [25, Theorem 7],

under the identification EndSHk(Sk) ∼= GW (k), we have χ(Spec(LH)) = trLH/k(〈1〉).

A field k is pythagorean if and only if sums of squares in k are squares in k. Since A(G) isalways torsion free as an abelian group, the importance of pythagorean fields in our contextis illustrated by the following lemma.

Lemma 3.2. The abelian group underlying GW (k) is torsion free if and only if the fieldk is pythagorean. If k is pythagorean with finitely many orderings, then the free rank ofGW (k) is 1 + x(k) where x(k) denotes the number of orderings of k.

Proof. This is a standard enhancement of [33, Theorem VIII.4.1 & Corollary VIII.6.15] fromthe Witt ring to Grothendieck-Witt ring case.

We will also need the following lemma in order to analyze hL/k.

Lemma 3.3. If k is pythagorean and [k×: (k×)2] = 2n, then

n ≤ x(k) ≤ 2n−1.

Proof. This is a specialization of [33, Exercise VIII.16].

Recall that k is euclidean if −1 is not a sum of squares in k and [k×: (k×)2] = 2.

Theorem 3.4. The map hL/k is an isomorphism if and only if either k is quadraticallyclosed and L = k, or k is euclidean and L = k[i].

Proof. If L/k is of one of the prescribed forms, then it is elementary that hL/k is an isomor-phism.

If hL/k is an isomorphism, then GW (k) must be torsion free, in which case Lemma 3.2implies that k is pythagorean. If k is pythagorean and nonreal (i.e., −1 is a sum of squaresin k), then k is quadratically closed and GW (k) ∼= Z. Thus A(G) has rank 1 and thereforeG = e and L = k.


Now assume that k is pythagorean and formally real (so −1 is not a sum of squares ink). By the construction in [3, §4], we know that h factors through the group completionof the monoid of k-quadratic forms q such that qL ∼= n〈1〉 for some natural number n; callthis group GW Z

L(k). Since h is an isomorphism, GW ZL(k) = GW (k), whence 〈a〉L = 〈1〉 in

GW (L) for all a ∈ k×. It follows that k is quadratically closed in L.Choose a basis x1, x2, . . . of k×/(k×)2 and let E = k(

√x1,

√x2, . . .). We have just

proven that E/k is a subextension of L/k, whence G surjects onto Gal(E/k). Since G isfinite, k must have finitely many square classes and Gal(E/k) ∼= Cn2 . Recall that the rankof A(G) is the number of conjugacy classes of subgroups of G. We deduce that rkA(G) ≥rkA(Cn2 ). Just counting the subgroups of Cn2 of order 1 or 2, we find that rkA(Cn2 ) ≥ 2n.Since 2n > 1 + 2n−1 for n > 2, Lemma 3.3 implies that n = 0 or 1. Since k is formally real,we can exclude the case n = 0, whence k is formally real pythagorean with [k×: (k×)2] = 2,i.e., k is euclidean. In this case GW (k) has rank 2, so L/k is a quadratic extension. Sincek is quadratically closed in L, L = k[i], concluding the proof.

Corollary 3.5. If c∗L/k is full and faithful, then k is of the form described in Theorem 3.4.

Remark 3.6. Algebraically closed and real closed fields are special examples of quadrati-cally closed and euclidean fields, but there are many other examples of these kinds of fields.

For instance, the field of real constructible numbers Q∩R (where Q is the quadratic closureof Q) is euclidean but not real closed.

The necessary conditions which we deduced in the previous result were obtained onlyby analyzing the zeroth homotopy group of the sphere spectrum. The authors expect thattorsion phenomena in the higher homotopy groups of Sk will preclude c∗L/k from being full

and faithful unless k is algebraically or real closed.

Conjecture 3.7. Let L be a field of characteristic zero. The functor c∗L/L : SH → SHL is

full and faithful if and only if L is algebraically closed.

By Levine’s theorem [34] the “if” portion of this conjecture is valid. Observe that thevalidity of this conjecture together with Corollary 2.26, would imply that cL/k : SHG → SHkis full and faithful if and only if k = L is algebraically closed or k is real closed and L = k[i].It is also interesting to ask what happens in positive characteristic.

4. Comparison functors

In this section we construct and analyze the various comparison functors between stablehomotopy categories used in our arguments. To avoid potential confusion concerning nota-tion, we point out that a functor on homotopy categories written as the derived functor LF(or RF ) of some functor on model categories in this section would be written simply F inprevious sections.

4.1. Motivic model structures. Given a base scheme S, the category Spc•(S) of basedmotivic spaces is the category of based simplicial presheaves on Sm/S. There are manydifferent options for a motivic model structure on Spc•(S). We will use the so-called closedflasque motivic model structure introduced in [43]. We recall the basic definitions below andrefer to loc. cit. for full details. The main advantages of this model structure for the presentwork are that in this model structure all of the standard motivic spheres are cofibrant andall of the various change of base functors as well as the (equivariant) Betti realizations areQuillen functors.


The closed flasque motivic model structure is a Bousfield localization of the global closedflasque model structure. The weak equivalences of the global closed flasque model structureare the schemewise weak equivalences of motivic spaces. A global closed flasque fibration is amap which has the right lifting property with respect to the set Jgf defined below. A globalclosed flasque cofibration is then defined by the appropriate lifting property. This modelstructure has sets of generating cofibrations Igf and generating acyclic cofibrations Jgf asfollows. Let Z = Zi → X be a finite (possibly empty) collection of closed immersions inSm/S. Write ∪Z for the categorical union (i.e. union as presheaves) of the Zi and writef : ∪Z → X for the induced map. Given two maps α and β write αβ for their pushoutproduct.

(1) The set Igf consists of all maps of the form f+ g+ where f : ∪Z → X is as aboveand g : ∂∆n → ∆n is a generating cofibration of simplicial sets.

(2) The set Jgf consists of all morphisms of the form f+ h+, where f : ∪Z → X is asabove and h : Λnj → ∆n is a generating acyclic cofibration.

The global closed flasque model structure is a proper, cellular, simplicial model structure.Write Spc•(S)gf for the category of motivic spaces equipped with the global model structure.Let

B //

Y

p

A

i // X.

be a distinguished Nisnevich square, i.e. p is an etale map of smooth schemes, i is an openimmersion, and p−1(X r A)red → (X r A)red is an isomorphism. Write Q = Q(i, p) forthis distinguished square and write PQ for the homotopy pushout in Spc•(S)gf of A andY along B, and write PQ → X for the resulting map. The motivic closed flasque modelstructure is the left Bousfield localization of the global model structure at the set of maps

S = PQ → X ∪ W × A1 →Wwhere X , W range over all smooth S-schemes and Q ranges over all distinguished squares.

4.2. Stable model structures. We rely on [23] as needed to equip various categories ofspectra (and bispectra) with stable model structures. Recall that if C is a left proper cellularsymmetric monoidal model category whose generating cofibrations have cofibrant domainand K is a cofibrant object of C, then Hovey equips the category SptΣK(C) of symmetricK-spectra with a stable model structure and it is again a left proper cellular symmetricmonoidal model category [23]. Note that Spc•(S) satisfies these assumptions and moreoverthe motivic spheres P1 (based at ∞), A1/A1 r 0, and Sα := A1 r 0 (based at 1) are allclosed flasque cofibrant.

Let J be a closed flasque cofibrant motivic space over S. We will simply write SptΣJ (S) :=

SptΣJ (Spc•(S)) for the category of motivic J-spectra. If J ′ is another closed flasque cofibrant

motivic space we write SptΣJ,J′(F ) := SptΣJ′(SptΣJ (F )) for the category of motivic (J, J ′)-

bispectra. As shown in [43] there is a monoidal Quillen equivalence between SptΣP1(S) andJardine’s model category of motivic symmetric P1-spectra [31].

In [23], functoriality of the model categories of symmetric spectra is discussed when Cis fixed (e.g. changing the suspension object K in C or varying the C-model category). Wewill need slightly more general functoriality, which we record before continuing with theconstruction of the comparison functors of interest to this paper.


Suppose that D is another model category satisfying the same hypothesis as C and K ′ isa cofibrant object of D. Further suppose that we are given the following:

(1) a Quillen adjoint pair Φ : C D : Ψ, and

(2) a natural isomorphism τ : Φ(−) ⊗K ′∼=−→ Φ(− ⊗K) such that the iterated isomor-

phisms τp : Φ(X) ⊗ (K ′)⊗p ∼= Φ(X ⊗K⊗p) are Σp-equivariant, where the actionsare the obvious ones given by permuting the respective factors of K and K ′.

As seen in the next lemma, (Φ,Ψ) prolong to a Quillen pair (Sp(Φ), Sp(Ψ)) of stablemodel categories of symmetric spectra. In this situation, we usually write Φ and Ψ insteadof Sp(Φ) and Sp(Ψ) for the prolongations.

Lemma 4.1. With notations and assumptions as above, the pair (Φ,Ψ) prolongs to aQuillen adjoint pair on stable model structures

Sp(Φ) : SptΣK(C) SptΣK′(D) : Sp(Ψ).

If Φ is strong symmetric monoidal then so is Sp(Φ).

Proof. Define Sp(Φ)(D) by Sp(Φ)(D)n := Φ(Dn) with structure maps

Sp(Φ)(D)n = Φ(Dn)⊗K ′ ∼= Φ(Dn ⊗K) → Φ(Dn+1) = Sp(Φ)(D)n+1.

The equivariance assumption on τ implies that the iterations of the structure mapSp(Φ)(D)n ⊗ (K ′)⊗p → Sp(Φ)(D)n+p are Σn ×Σp-equivariant and so Φ(D) is a symmetricK ′-spectrum. Define Sp(Φ) on morphisms in the obvious way.

Note that τ determines the natural isomorphism ρ : ΨΩK′(−)∼=−→ ΩKΨ(−) and the

iterations ρp are Σp-equivariant. Now define Sp(Ψ)(E) by setting Sp(Ψ)(E)n := Ψ(En).The structure maps are defined as the adjoints of

Sp(Ψ)(E)n = Ψ(En) → Ψ(ΩK′En+1) ∼= ΩKΨ(En+1) = ΩKSp(Ψ)(E)n+1.

The equivariance of ρ implies that this is a symmetric K-spectrum. Define Sp(Ψ) on mor-phisms in the obvious way.

It is straightforward to verify that Sp(Φ) and Sp(Ψ) are adjoint. The functor Sp(Ψ) pre-serves level equivalences and level fibrations. This implies Sp(Φ) preserves stable cofibrationsand Sp(Ψ) preserves fibrations between fibrant objects in the stable model structure. It fol-lows from [11, Lemma A.2] that (Sp(Φ), Sp(Ψ)) is a Quillen adjoint pair on the stable modelstructures.

It is immediate that Sp(Φ) is symmetric monoidal whenever Φ is.

4.3. Galois correspondence. Let L/k be a finite Galois extension with Galois group G.Define the functor

(4.2) cL/k : OrG → Sm/k,

by cL/k(G/H) = Spec(LH) on objects and on maps as follows. First recall that

HomOrG(G/H,G/H′) = gH ′ | g−1Hg ⊆ H ′.

A straightforward check shows that if gH ′ is such a coset then the corresponding fieldautomorphism g : L → L restricts to a map of fields g : LH

′ → LH which depends only onthe coset gH ′. This defines the desired map cL/k(G/H) → cL/k(G/H

′).The category of G-simplicial sets is equivalent to the category of presheaves of simplicial

sets on OrG: the presheaf corresponding to A is given by G/H 7→ AH . We thus obtain anadjoint pair of functors

(4.3) c∗L/k : GsSet• Spc•(k) : (cL/k)∗.


Remark 4.4. For a G-simplicial set A, the corresponding motivic space c∗L/k(A) isn’t

in general constant but its possible values are limited to the various fixed points AH forsubgroups H ⊆ G. To see this, it suffices to consider the case of a G-set. Every G-set isthe disjoint union of orbits and we write this decomposition as A =

∐OrG

∐A〈G/H〉G/H .

Then c∗L/kA is the motivic space defined by

(c∗L/kA)(X) :=∐

OrG

∐

A〈G/H〉

Homk(X, Spec(LH)).

Note that if X is connected, then Homk(X, Spec(LH)) is either empty or is a set with |G/H |

elements and so c∗L/k(A)(X) = AH for an appropriate subgroup H ⊆ G.

Lemma 4.5. The adjoint pair c∗L/k : GsSet• Spc•(k) : (cL/k)∗ is a Quillen adjoint pair.

Moreover, the induced map on homotopy categories Lc∗L/k : H•,G → H•,k is full and faithful.

Proof. Note that under the identification sPre•(OrG) = GsSet•, the projective model struc-ture on simplicial presheaves corresponds to the usual model structure on based G-simplicialsets. The functor (cL/k)∗ preserves global weak equivalences and global fibrations and so thispair is a Quillen pair on the global closed flasque model structure. It follows immediatelythat this is a Quillen pair on the motivic model structure as well.

Using the description of c∗L/k in the previous remark, one sees the following simple facts

about c∗L/k(A). If A is fibrant then c∗L/k(A)(X) is fibrant for any X and c∗L/k(A) is A1-

homotopy invariant. If U ⊆ X is a dense open subscheme, then c∗L/k(A)(X) = c∗L/k(A)(U).

It is thus easy to see that c∗L/k(A) satisfies Nisnevich descent. Moreover, for G-simplicial sets

A and B, we have an equality of simplicial mapping spaces, HomSpc•(k)(c

∗L/k(B), c∗L/k(A)) =

HomGsSet•(B,A). Now if B is cofibrant and A is fibrant, then we have

[Sn ∧ c∗L/k(B), c∗L/k(A)]k = πnHomSpc•(k)(c

∗L/k(B), c∗L/k(A))

and [B,A]G = πnHomGsSet•(B,A) from which the second statement follows.

Write SG = (S1)∧G for the G-simplicial set consisting of the |G|-fold smash product ofS1 equipped with the obvious permutation action by G. Note also that this is the simpli-cial representation sphere associated to the regular representation of G. The stable modelstructure on SptΣSG(G) := SptΣSG(GsSet•) obtained from [23] agrees with that constructed in[36]. In turn, as shown in loc. cit., the associated homotopy category is tensor triangulatedequivalent to the genuine G-equivariant homotopy category as constructed in [35].

To simplify notation below, we sometimes denote the motivic space c∗L/k(SG) by SG.

Consider the category SptΣSG,P1(k) of motivic (c∗L/k(SG),P1

k)-bispectra. This is a model for

the stable motivic homotopy category SHk. Indeed, by [26, Theorem 3.5] the motivic spacec∗L/k(S

G) is invertible in SHk. In particular, by [23, Theorem 9.1], the suspension spectrum

functorΣ∞SG : SptΣP1(k) → SptΣP1,SG(k)

is a left Quillen equivalence and induces a tensor triangulated equivalence on the associatedstable homotopy categories.

By Lemma 4.1, the Quillen adjoint pair (4.3) induces a Quillen pair SptΣSG(G) SptΣSG(k).Combined with the suspension spectrum functor, we have the composite Quillen adjunction

SptΣSG(G) SptΣSG(k) SptΣSG,P1(k).

We have thus obtained the desired stabilization of c∗L/k.


Theorem 4.6. The Galois correspondence (4.2) induces an adjoint pair

Lc∗L/k : SHG SHk : R(cL/k)∗

of triangulated stable homotopy categories. The left adjoint is strong symmetric monoidal.

4.4. Equivariant Betti realization. An unstable C2-equivariant Betti realization functoris constructed for the motivic homotopy category over fields admitting a real embedding in[40], see also [13]. It is well known that this construction stabilizes to yield a C2-equivariantBetti realization functor. Following the construction of [43] in the complex case, we recordhere the construction of the stable equivariant Betti realization as a Quillen functor.

Write (−)an : Sm/R → C2Top• for the functor given by X 7→ X(C)an+ , where X(C) isequipped with the involution given by conjugation. It extends to an adjoint pair

ReC2

B : Spc•(R) C2Top• : SingC2

B .

The left adjoint ReC2

B is defined by the usual left Kan extension formula and the right adjoint

SingC2

B is defined by SingC2

B (K)(X) = HomC2Top•

(X(C)+,K).

Proposition 4.7. The adjoint pair ReC2

B : Spc•(R) C2Top• : SingC2

B is a Quillen adjoint

pair. Moreover ReC2

B is strong symmetric monoidal.

Proof. First we show that this is a Quillen pair on global closed flasque model structures.For this we check that ReC2

B sends generating closed cofibrations to cofibrations in C2Top•and sends generating global trivial closed fibrations to trivial cofibrations. Note that ReC2

B

preserves pushout products. It thus suffices to show that ReC2

B (∪Z+) → ReC2

B (X+) is acofibration for any finite collection Z = Zi → X of closed immersions in Sm/R.

Note that ReC2

B (∪Z) is the coequalizer of∐Zi(C)×X(C)Zj(C) ⇒

∐Zi(C) in in C2Top•.

One may equivariantly triangulate X(C) such that each Zi(C) is an equivariant subcomplexand Zi(C) ×X(C) Zj(C) is an equivariant subcomplex for each j, see, e.g., [30]. It follows

that ReC2

B (∪Z) → X(C) is the inclusion of an equivariant subcomplex. In particular, it is

an equivariant cofibration. It follows that ReC2

B is a left Quillen functor on the global closedflasque model structure.

Note that ReC2

B sends a distinguished Nisnevich square to an equivariant homotopy

pushout square, see, e.g., [13]. Also ReC2

B (X × A1) → ReC2

B (X) is an equivariant homo-topy equivalence. It follows that the adjoint pair of the proposition induces a Quillen pairin the closed flasque motivic structure as well.

Recall that we write Sσ for the sign representation sphere.

Proposition 4.8. The above adjoint pair extends to a Quillen adjoint pair

ReC2

B : SptΣP1(R) SptΣS1+σ(C2) : SingC2

B

on stable model categories. Moreover ReC2

B is strong symmetric monoidal.

Proof. This follows immediately from Lemma 4.1, noting that ReC2

B (P1) = S1+σ.

Now if k is a field and φ : k → R is a real embedding then the associated C2-equivariantBetti realization ReC2

B,φ is defined to be the composite

ReC2

B,φ := φ∗ ReC2

B : SHk → SHR → SHC2.


4.5. Comparing change of group and change of base functors. It is useful to knowthat the comparison functors between equivariant and motivic homotopy theory suitablyintertwine the standard change of group and change of base functors. We fix as above aGalois extension L/k with Galois group G. Let H ⊆ G be a subgroup and write K = LH

for the corresponding fixed subfield. We denote the corresponding map of schemes byp : Spec(K) → Spec(k). As with any map of schemes we have an induced adjoint pair offunctors of motivic spaces p∗ : Spc•(k) Spc•(K) : p∗. Since p is smooth, the functor p∗

has as well a left adjoint p#, induced by the functor Sm/K → Sm/k which composes thestructure map of a K-scheme with p.

Lemma 4.9. The adjoint pairs (p#, p∗) and (p∗, p∗) are Quillen adjoint pairs.

Proof. That p∗ is a left Quillen adjoint on the motivic closed flasque model structure isverified in [43]. Note that p# preserves generating global closed flasque cofibrations andacyclic cofibrations. This is seen by noting that if M is a simplicial set, then we have thatp#(X ∧M+) = (p#X) ∧M+ and since p# preserves colimits, it preserves pushouts andsince it also preserves closed inclusions of smooth schemes, the claim follows. This impliesthat p# is a left Quillen functor on global closed flasque model structures. The functorp# sends Nisnevich distinguished squares to Nisnevich distinguished squares. Furthermorep#(X ×K A1

K) → p#(X) is identified with p#(X) ×k A1k → p#(X). It follows that p# is

also a left Quillen functor on the closed flasque motivic model structure.

We have the commutative diagram of categories

OrH

j

cL/K // Sm/K

p#

OrG

cL/k // Sm/k,

where j sends the orbit H/H ′ to the orbit G/H ′. Under the identification sPre•(OrG) =

GsSet•, the adjoint pair (j∗, j∗) is identified with the adjoint pair (indGH , resGH) where

indGH(X) = G ×H X and resGH(W ) is W with H-action given by restricting the G-action.The above square thus induces a commutative diagram of Quillen adjoint functors (wherewe omit the labels for the horizontal right adjoints for typographical reasons)

(4.10) HsSet•

c∗L/K //

indGH

Spc•(K)

p#

oo

GsSet•

resGH

OO

c∗L/k //Spc•(k).

p∗

OO

oo

We write H•,G for the homotopy category of based G-spaces and H•,k for the unstablemotivic homotopy category.

Proposition 4.11. The diagrams of homotopy categories

H•,H

Lc∗L/K // H•,K

H•,G

RresGH

OO

Lc∗L/k // H•,k

Rp∗

OOand H•,H

LindGH

Lc∗L/K // H•,K

Lp#

H•,G

Lc∗L/k // H•,k.

induced by (4.10), commute up to natural isomorphism.


Proof. The commutativity of the second diagram follows immediately from the fact that(4.10) commutes and that the adjoint pairs there are Quillen pairs. A direct inspection yieldsthe equality of functors p∗c∗L/k = c∗L/KresGH . The commutativity of the first diagram follows

since p∗ and resGH are also left Quillen functors and so Rp∗ ≃ Lp∗ and RresGH ≃ LresGH .

As an H-simplicial set SG is isomorphic to the [G : H ]-fold smash product of SH . Thisimplies that c∗L/K(SG) = c∗L/k(S

H)∧[G:H]. We set d := [G : H ] and write SdH = (SH)∧d

below.

Proposition 4.12. The adjoint pairs (4.10) induce diagrams of stable homotopy categories

SHHLc∗L/K // SHK

SHG

RresGH

OO

Lc∗L/k // SHk

Rp∗

OO and SHH

LindGH

Lc∗L/K // SHK

Lp#

SHG

Lc∗L/k // SHk

which commute up to natural isomorphism.

Proof. We have a diagram of model categories and Quillen adjunctions between them

SptΣSG(H)

indGH

c∗L/K //SptΣSdH (K)

p#

ooΣ∞

P1 //SptΣSdH ,P1(K)

p#

oo

SptΣSG(G)c∗L/k //

resGH

OO

SptΣSG(k)Σ∞

P1 //oo

p∗

OO

SptΣSG,P1(k).

p∗

OO

oo

This diagram is commutative and the derived functors of the left adjoints give the functorsin the diagrams. The commutativity of the second diagram follows immediately.

For the commutativity of the first square, note that the right adjoints p∗ and resGH arealso left adjoints and the stabilization of these functors considered as a left adjoint agreeswith their stabilization as a right adjoint and these are also left Quillen functors. It followsthat Rp∗ = Lp∗ and RresGH = LresGH . The desired commutativity thus follows from theunderived equality Σ∞

P1c∗L/KresGH = p∗Σ∞P1c∗L/k.

Now suppose that k is formally real and consider the embedding p : k ⊆ k[i]. A realembedding φ : k → R induces a complex embedding ψ : k[i] → C and hence an associatedBetti realization ReB,ψ = ψ∗ReB : SHk[i] → SH.

Proposition 4.13. With the notations as above we have

RresC2

eLReC2

B,φ = LReB,ψRp∗ and LReC2

B,φLp# = LindC2

eLReB,ψ.

Proof. This is a straightforward consequence of the definitions and constructions, as in theprevious proposition.

4.6. Betti realization and motivic cohomology. We now turn our attention to the equi-variant Betti realization of the motivic cohomology spectrum. Following a similar strategyas in [34] in the nonequivariant case, we show that the equivariant Betti realization takesthe motivic cohomology spectrum HZ to the Bredon cohomology spectrum HZ. We thenreinterpret the Beilinson-Lichtenbaum conjectures and establish an equivariant version ofSuslin-Voevodsky’s theorem [46] on Suslin homology.


Lemma 4.14. For any X in Sch/k, the natural map LReC2

B,φ(Σ∞P1X+) → Σ∞

S1+σX(C)an+ isan isomorphism in SHC2

.

Proof. Since k admits resolution of singularities there is a proper cdh hypercover X• → Xsuch that each Xn is a smooth k-scheme. It follows from [51] that |Σ∞

P1X•+| → Σ∞P1X+

is a stable equivalence in SHk. Each Xn+ is cofibrant. It follows that we have a natural

isomorphism LReC2

B,φΣ∞P1X+

∼= |Σ∞S1+σX(C)•+| in SHC2

.

To see that |Σ∞S1+σX(C)an•+| → Σ∞

S1+σX(C)an+ is an isomorphism in SHC2it suffices to

check that this map induces an isomorphism in SH after applying the geometric fixedpoints functors ΦC2 and Φe. Recall that in general we have that the geometric fixedpoints of a suspension spectrum is given by the suspension spectrum of the fixed points:ΦHΣ∞

SGY = Σ∞Y H . Therefore we have that the C2-geometric fixed points of the abovemap is |Σ∞X(R)an•+| → Σ∞X(R)an+ . If W → Y is a proper cdh-cover of real varietiesthen W (R)an → Y (R)an is a surjective proper map. In particular, it is a map of universalcohomological descent [7, 5.3.5]. It follows that H∗(|X(R)an•+|, A) → H∗(X(R)an+ , A) is anisomorphism for all abelian groups A. In particular, |X(R)an•+| → X(R)an+ induces a stableequivalence on suspension spectra. A similar analysis for the e-geometric fixed points showsthat |Σ∞X(C)an•+| → Σ∞X(C)an+ is a stable equivalence as well.

Lemma 4.15. The natural map

LReC2

B,φ(Σ∞P1 Sym

N (ΣmP1Y+)) → Σ∞S1+σ Sym

N (ΣmS1+σY (C)an+ )

is an isomorphism in SHC2for any N , m and any Y in Sm/k.

Proof. The argument is identical to [34, Lemma 5.4]. The key point is that there is ahomotopy pushout square in Spc•(k) of the form

SymN (X,A) //

SymN (X)

SymN−1(Σm

P1Y+) // SymN (ΣmP1Y+)

where X = (P1)m × Y+ and A is the closed subscheme of points (x1, . . . , xm, y) such thatsome xi = ∞. The previous lemma applied to the top two vertices and induction on Napplied to the lower left vertex yields the result.

As in [34] we write

(Σ∞P1X+)

treff := (Sym∞X+, Sym

∞(ΣP1X+), Sym∞(Σ2

P1X+), . . .)

and together with the obvious structure maps. Similarly for a C2-space W we have the C2-spectrum (Σ∞

S1+σW+)treff := Sym∞(ΣmS1+σW+)m≥0, equipped with the obvious structure

maps.

Proposition 4.16. For any smooth X there is a natural isomorphism in SHC2

LReC2

B,φ(Σ∞P1X+)

treff

∼= (Σ∞S1+σX(C)an+ )treff .

Proof. We have the natural isomorphism colimn(Σ∞P1En)[n] ∼= E in SHk, where D[n] is the

shifted spectrum given by (D[n])i = Di−n. Similarly we have the natural isomorphism

colimn(Σ∞S1+σFn)[n] ∼= F in SHC2

. Since LReC2

B,φ preserves homotopy colimits and shifts,the result follows from the previous lemma.


Theorem 4.17. Let Λ be an abelian group. There is an isomorphism in SHC2

LReC2

B,φ(HΛ)∼= HΛ.

Proof. Since HΛ = HZ ∧ MΛ and HΛ = HZ ∧ MΛ, where MΛ is a Moore spectrum forΛ, and LReC2

B,φ(MΛ) = MΛ, it suffices to establish the result for A = Z. The motivic

cohomology spectrum HZ is given by HZn = Ztr((P1)∧n) and equipped with the obviousstructure maps. The natural map (Sk)

treff → HZ is an isomorphism in SHk by [34, Lemma

5.9]. It follows from [9, Proposition 3.7] that the spectrum ZSn(1+σ)n≥0 is a model forHZ, i.e. it represents Bredon cohomology with coefficients in the constant Mackey functorZ. It follows from [12, Corollary A.7] that the natural map (SC2

)treff → ZSn(1+σ)n≥0 is

an equivariant weak equivalence. By the previous proposition, LReC2

B,φ((Sk)treff ) = (SC2

)treffand the result follows.

The Beilinson-Lichtenbaum conjectures assert that for any smooth variety X over a fieldk, any n > 1, and q ≥ 0, the generalized cycle map

Hp+qαM (X,Z/n) → Hp+q

et (X,µ⊗qn )

is an isomorphism for p ≤ 0 and an injection for p = 1. By a theorem of Suslin-Voevodsky[47], these conjectures are equivalent to the Bloch-Kato conjectures. In turn, these havebeen resolved by Voevodsky in case n = 2ℓ and in general by Voevodsky and Rost. Supposenow that k = R. The etale cohomology (with finite coefficients) of the real variety X can be

identified with the Borel cohomology ofX(C). On the other hand ReC2

B induces a comparisonmap between motivic cohomology and Bredon cohomology and we would like to reinterpretthe Beilinson-Lichtenbaum conjectures as a statement concerning this comparison. When 2is invertible in the coefficient group this is straightforward. In [21] the first author and M.Voineagu treat the case of coefficient group Z/2ℓ by carefully comparing various cycle mapstogether with a computation that Bredon and Borel cohomology agree in the appropriaterange. This reinterpretation of Voevodsky’s theorem applies more generally to the Bettirealization for an embedding of a real closed field into R.

Theorem 4.18. Let φ : k → R be an embedding with k real closed and X a smooth k-variety. For any n ≥ 1, and any q ≥ 0 the map

Hs+qαM (X,Z/n) → Hs+qσ(X(C),Z/n),

induced by ReC2

B,φ, is an isomorphism for s ≤ 0 and an injection for s = 1.

Proof. Motivic cohomology forms a pretheory with transfers. Applying [45, Theorem 1],6

we have that the base change φ∗ : Hs+qαM (X,Z/n) → Hs+qα

M (XR,Z/n) is an isomorphism soit suffices to treat the case k = R.

6This rigidity result is stated for dense subfields of a henselian valued field. Unfortunately R can’t beequipped with a nontrivial henselian valuation. However, the proof of their result relies only on the densitylemma [45, Lemma 1] which is valid for a real closed subfield of R, with the classical topology. This iswell-known, see e.g. [32, Lemma 4] for a proof.


Suppose that 2 is invertible in Z/n and write p : Spec(C) → Spec(R) for the canonicalmap. Using Proposition 4.13 we have the commutative diagram induced by XC → X

Hs+qαM (X,Z/n)

p∗ //

ReC2B

Hs+qαM (XC,Z/n)

p# //

ReB

Hs+qαM (X,Z/n)

ReC2B

Hs+qσ(X(C),Z/n) // Hs+q

sing (X(C),Z/n) // Hs+qσ(X(C),Z/n).

The middle arrow is an isomorphism for s ≤ 0 and an injection for s = 1. The horizontalmaps are multiplication by 2, hence isomorphisms. The result thus follows for coefficientgroups in which 2 is invertible.

It remains to treat the case Z/2ℓ. The cycle map Hs+qαM (X,Z/2ℓ) → Hs+qσ(X(C),Z/2ℓ)

considered in [21] is induced by the map of simplicial abelian groups (for q ≥ 0)

HomR(X ×∆•R, Sym

∞ Pn)+

HomR(X ×∆•R, Sym

∞ Pn−1)+→ HomC2Top•

((X(C)×∆•top)+,Z(S

n(1+σ)))

obtained by sending an algebraic map of real varieties to its associated equivariant contin-uous map of C2-spaces. This agrees with the map considered here. By [21, Theorem 1.5,Proposition 5.1] it induces an isomorphism for s ≤ 0 and an injection for s = 1.

We finish with an equivariant version of Suslin-Voevodsky’s theorem [46] that over analgebraically closed field Suslin homology agrees with etale homology. To set the stage,fix a real embedding φ : k → R and consider the subcategory of motivic spectra X suchthat LReC2

B,φ induces an isomorphism [Sn, X ]k ∼= [Sn,LReC2

B,φ(X)]C2for all n. This is a

localizing subcategory of SHk and we show that it contains all effective torsion motives. Ifthe motivic slice tower were convergent we would be able to show more generally that itcontains all effective torsion motivic spectra (i.e. the localizing subcategory generated byΣsS1ΣtP1Σ∞

P1X/N for any s ∈ Z, t ≥ 0, N > 1, and smooth X).

Theorem 4.19. Let k be a real closed field and φ : k → R be an embedding. Let E be inthe smallest localizing subcategory of SHk containing X+ ∧ HZ/r for any smooth projectiveX and r > 1. Then for any n, the equivariant Betti realization induces an isomorphism

ReC2

B,φ : [Sn, E]k∼=−→ [Sn,ReC2

B,φ(E)]C2.

Proof. It suffices to show that [Sn, X ∧HZ/r]k → [Sn, XR(C)∧HZ/r]C2is an isomorphism

for any smooth projective X . As in the previous theorem, using [45, Theorem 1], we arereduced to the case k = R. Tracing through definitions, it suffices to show that the map

Ztr(X)(∆•R)⊗ Z/r = HomR(∆

•R, Sym

∞X)+ ⊗ Z/r → HomC2Top•(∆•

top,ZX(C))⊗ Z/r

of simplicial abelian groups, obtained by sending an algebraic map of real varieties to itsassociated equivariant continuous map of C2-spaces, is a homotopy equivalence. Note thatthis last simplicial abelian group equals Sing•(ZX(C))C2 ⊗ Z/r.

That this map is a homotopy equivalence can be deduced by a variant of some argumentsof Friedlander-Walker [19] as follows. First, for a presheaf F on Sch/R, define F (∆d

top) =colim∆d

top→W (R) F (W ) where the colimit ranges over continuous maps and W a finite type

real variety. Note that if F is the presheaf represented by a real variety Y then F (∆•top) =

Sing• Y (R). Consider the presheaf of simplicial abelian groups

G(−) := Ztr(X)(−×∆•R)⊗ Z/r.


Note that G(∆•top) and [HomC2Top•

(∆•top, Sym

∞X(C))]+ ⊗ Z/r are naturally homotopic.Combined with Quillen’s theorem [18, Appendix Q] on homotopy group completions ofsimplicial abelian monoids and the fact that (ZX(C))C2 is the homotopy group completion of(NX(C))C2 , we find that there is a natural homotopy equivalence of simplicial abelian groupsG(∆•

top) ≃ Sing•(ZX(C))C2 ⊗ Z/r. It thus suffices to show that the map G(R) → G(∆•top)

(induced by the projections ∆dtop → ∗) is a homotopy equivalence. This is easily seen via

the same argument as in [21, Proposition 5.1].

References

[1] J. F. Adams. Stable homotopy and generalised homology. University of Chicago Press, Chicago, Ill.,1974. Chicago Lectures in Mathematics.

[2] S. Araki and K. Iriye. Equivariant stable homotopy groups of spheres with involutions. I. Osaka J.Math., 19(1):1–55, 1982.

[3] P. W. Beaulieu and T. C. Palfrey. The Galois number. Math. Ann., 309(1):81–96, 1997.[4] M. Bokstedt, W. C. Hsiang, and I. Madsen. The cyclotomic trace and algebraic K-theory of spaces.

Invent. Math., 111(3):465–539, 1993.[5] A. K. Bousfield. The localization of spectra with respect to homology. Topology, 18(4):257–281, 1979.[6] D.-C. Cisinski and F. Deglise. Triangulated categories of mixed motives. ArXiv e-prints, 0912.2110v3.

[7] P. Deligne. Theorie de Hodge. III. Inst. Hautes Etudes Sci. Publ. Math., (44):5–77, 1974.[8] E. S. Devinatz and M. J. Hopkins. Homotopy fixed point spectra for closed subgroups of the Morava

stabilizer groups. Topology, 43(1):1–47, 2004.[9] P. F. dos Santos. A note on the equivariant Dold-Thom theorem. J. Pure Appl. Algebra, 183(1-3):299–

312, 2003.[10] A. W. M. Dress. Notes on the theory of representations of finite groups. Universitat Bielefeld, Fakultat

fur Mathematik, Bielefeld, 1971.[11] D. Dugger. Replacing model categories with simplicial ones. Trans. Amer. Math. Soc., 353(12):5003–

5027 (electronic), 2001.[12] D. Dugger. An Atiyah-Hirzebruch spectral sequence for KR-theory. K-Theory, 35(3-4):213–256 (2006),

2005.[13] D. Dugger and D. Isaksen. Topological hypercovers and A1-realizations. Math. Z., 246(4):667–689, 2004.[14] Daniel Dugger and Daniel C. Isaksen. The motivic Adams spectral sequence. Geom. Topol., 14(2):967–

1014, 2010.

[15] W. G. Dwyer and E. M. Friedlander. Algebraic and etale K-theory. Trans. Amer. Math. Soc.,292(1):247–280, 1985.

[16] H. Fausk. Equivariant homotopy theory for pro-spectra. Geom. Topol., 12(1):103–176, 2008.[17] H. Fausk, P. Hu, and J. P. May. Isomorphisms between left and right adjoints. Theory Appl. Categ.,

11:No. 4, 107–131, 2003.[18] E. Friedlander and B. Mazur. Filtrations on the homology of algebraic varieties. Mem. Amer. Math.

Soc., 110(529):x+110, 1994. With an appendix by Daniel Quillen.[19] E. Friedlander and M. Walker. Rational isomorphisms between K-theories and cohomology theories.

Invent. Math., 154(1):1–61, 2003.[20] J. P. C. Greenlees and J. P. May. Generalized Tate cohomology. Mem. Amer. Math. Soc.,

113(543):viii+178, 1995.[21] J. Heller and M. Voineagu. Vanishing theorems for real algebraic cycles. Amer. J. Math., 134(3):649–

709, 2012.[22] M. Hill, M. Hopkins, and D. Ravenel. On the non-existence of elements of kervair invariant one. ArXiv

e-prints, 0908.3724v2.[23] M. Hovey. Spectra and symmetric spectra in general model categories. J. Pure Appl. Algebra, 165(1):63–

127, 2001.[24] M. Hoyois. From algebraic cobordism to motivic cohomology. ArXiv e-prints, 1210.7182v4.[25] M. Hoyois. Traces and fixed points in stable motivic homotopy theory. ArXiv e-prints, 1309.6147v1.[26] P. Hu. Base change functors in the A1-stable homotopy category. Homology Homotopy Appl., 3(2):417–

451, 2001.[27] P. Hu and I. Kriz. Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral

sequence. Topology, 40(2):317–399, 2001.






[28] P. Hu, I. Kriz, and K. Ormsby. Convergence of the motivic Adams spectral sequence. J. K-Theory,7(3):573–596, 2011.

[29] P. Hu, I. Kriz, and K. Ormsby. The homotopy limit problem for Hermitian K-theory, equivariant motivichomotopy theory and motivic Real cobordism. Adv. Math., 228(1):434–480, 2011.

[30] S. Illman. Smooth equivariant triangulations of G-manifolds for G a finite group. Math. Ann.,233(3):199–220, 1978.

[31] J. F. Jardine. Motivic symmetric spectra. Doc. Math., 5:445–553 (electronic), 2000.[32] K. Kato and S. Saito. Unramified class field theory of arithmetical surfaces. Ann. of Math. (2),

118(2):241–275, 1983.[33] T. Y. Lam. Introduction to quadratic forms over fields, volume 67 of Graduate Studies in Mathematics.

American Mathematical Society, Providence, RI, 2005.[34] M. Levine. A comparison of motivic and classical homotopy theories. ArXiv e-prints, 1201.0283v4.[35] L. G. Lewis, Jr., J. P. May, M. Steinberger, and J. E. McClure. Equivariant stable homotopy theory,

volume 1213 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. With contributions by J.E. McClure.

[36] M. Mandell. Equivariant symmetric spectra. In Homotopy theory: relations with algebraic geometry,group cohomology, and algebraic K-theory, volume 346 of Contemp. Math., pages 399–452. Amer.Math. Soc., Providence, RI, 2004.

[37] J. P. May. Picard groups, Grothendieck rings, and Burnside rings of categories. Adv. Math., 163(1):1–16,2001.

[38] F. Morel. Rational stable splitting of grassmannians and rational motivic sphere spectrum. preprint.[39] F. Morel. On the motivic π0 of the sphere spectrum. In Axiomatic, enriched and motivic homotopy

theory, volume 131 of NATO Sci. Ser. II Math. Phys. Chem., pages 219–260. Kluwer Acad. Publ.,Dordrecht, 2004.

[40] F. Morel and V. Voevodsky. A1-homotopy theory of schemes. Inst. Hautes Etudes Sci. Publ. Math.,(90):45–143 (2001), 1999.

[41] D. Orlov, A. Vishik, and V. Voevodsky. An exact sequence for KM∗

/2 with applications to quadraticforms. Ann. of Math. (2), 165(1):1–13, 2007.

[42] K. Ormsby and P. A. Østvær. Stable motivic π1 of low-dimensional fields. ArXiv eprints, 1310.2970v1.[43] I. Panin, K. Pimenov, and O. Rondigs. On Voevodsky’s algebraic K-theory spectrum. In Algebraic

topology, volume 4 of Abel Symp., pages 279–330. Springer, Berlin, 2009.[44] G. Quick. Profinite G-spectra. Homology Homotopy Appl., 15(1):151–189, 2013.[45] A. Rosenschon and P. A. Østvær. Rigidity for pseudo pretheories. Invent. Math., 166(1):95–102, 2006.[46] A. Suslin and V. Voevodsky. Singular homology of abstract algebraic varieties. Invent. Math., 123(1):61–

94, 1996.[47] A. Suslin and V. Voevodsky. Bloch-Kato conjecture and motivic cohomology with finite coefficients. In

The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), volume 548 of NATO Sci. Ser. CMath. Phys. Sci., pages 117–189. Kluwer Acad. Publ., Dordrecht, 2000.

[48] V. Voevodsky. A1-homotopy theory. In Proceedings of the International Congress of Mathematicians,Vol. I (Berlin, 1998), number Extra Vol. I, pages 579–604 (electronic), 1998.

[49] V. Voevodsky. Motivic cohomology with Z/2-coefficients. Publ. Math. Inst. Hautes Etudes Sci., (98):59–104, 2003.

[50] V. Voevodsky. Reduced power operations in motivic cohomology. Publ. Math. Inst. Hautes Etudes Sci.,(98):1–57, 2003.

[51] V. Voevodsky. Unstable motivic homotopy categories in Nisnevich and cdh-topologies. J. Pure Appl.Algebra, 214(8):1399–1406, 2010.

[52] V. Voevodsky. On motivic cohomology with Z/l-coefficients. Ann. of Math. (2), 174(1):401–438, 2011.[53] V. Voevodsky, A. Suslin, and E. Friedlander. Cycles, transfers, and motivic homology theories, volume

143 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2000.

University of Illinois Urbana-Champaign

E-mail address: [email protected]

Reed College

E-mail address: [email protected]



Documents

GALOIS EQUIVARIANCE AND STABLE MOTIVIC HOMOTOPY …Hopkins [8], stable equivariant homotopy theory controls the chromatic decomposition of stable homotopy theory. It is also essential